The Annals of Applied Probability

Reflected BSDEs when the obstacle is not right-continuous and optimal stopping

Miryana Grigorova, Peter Imkeller, Elias Offen, Youssef Ouknine, and Marie-Claire Quenez

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Abstract

In the first part of the paper, we study reflected backward stochastic differential equations (RBSDEs) with lower obstacle which is assumed to be right upper-semicontinuous but not necessarily right-continuous. We prove existence and uniqueness of the solutions to such RBSDEs in appropriate Banach spaces. The result is established by using some results from optimal stopping theory, some tools from the general theory of processes such as Mertens’ decomposition of optional strong supermartingales, as well as an appropriate generalization of Itô’s formula due to Gal’chouk and Lenglart. In the second part of the paper, we provide some links between the RBSDE studied in the first part and an optimal stopping problem in which the risk of a financial position $\xi$ is assessed by an $f$-conditional expectation $\mathcal{E}^{f}(\cdot)$ (where $f$ is a Lipschitz driver). We characterize the “value function” of the problem in terms of the solution to our RBSDE. Under an additional assumption of left upper-semicontinuity along stopping times on $\xi$, we show the existence of an optimal stopping time. We also provide a generalization of Mertens’ decomposition to the case of strong $\mathcal{E}^{f}$-supermartingales.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 5 (2017), 3153-3188.

Dates
Received: July 2015
Revised: August 2016
First available in Project Euclid: 3 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1509696043

Digital Object Identifier
doi:10.1214/17-AAP1278

Mathematical Reviews number (MathSciNet)
MR3719955

Zentralblatt MATH identifier
1379.60045

Subjects
Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 93E20: Optimal stochastic control 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 60G07: General theory of processes 47N10: Applications in optimization, convex analysis, mathematical programming, economics

Keywords
Backward stochastic differential equation reflected backward stochastic differential equation optimal stopping $f$-expectation strong optional supermartingale Mertens’ decomposition dynamic risk measure strong $\mathcal{E}^{f}$-supermartingale

Citation

Grigorova, Miryana; Imkeller, Peter; Offen, Elias; Ouknine, Youssef; Quenez, Marie-Claire. Reflected BSDEs when the obstacle is not right-continuous and optimal stopping. Ann. Appl. Probab. 27 (2017), no. 5, 3153--3188. doi:10.1214/17-AAP1278. https://projecteuclid.org/euclid.aoap/1509696043


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